Minimax estimation of the mode of functional data

Abstract

We give a definition of the mode of a probability distribution defined on an infinite-dimensional space and establish the minimax convergence rate. Therein, the existence of a probability density with respect to some abstract reference measure will not be required, while instead we use small ball probabilities to define the mode. The optimal rates of convergence are of logarithmic type under additional constraints on the small ball probabilities and entropy. We assume that the mode is contained in some totally bounded set and use covering methods to define a mode estimator and deduce strong consistency and rate optimality.